1. Reference problem#

1.1. Geometry#

Geometric values are expressed in meters.

Straight beam of length \(L=1\), direction \(x\).

_images/100000000000032E000000B10B745D3C987624CE.png

Two types of section are calculated simultaneously:

Rectangular section

_images/10000000000001060000011BC8363945E0463E5B.png

Circular section

_images/1000000000000074000000732A48C99847FF9367.png

For modeling D, 1 thin tube section is calculated:

_images/100000000000007800000078436C191CC3DB6F4F.png

1.2. Material properties#

\(E=2.{10}^{11}\mathrm{Pa}\)

\(\nu =0.3\)

ECRO_LINE:

\(\mathrm{SY}={\sigma }_{y}=150{10}^{6}\mathrm{Pa}\)

\(H\) = D_ SIGM_EPSI = \(2{10}^{9}\mathrm{Pa}\) or 0

1.3. Boundary conditions and loads#

Embedding in \(O\)

Displacement imposed in \(B\)

\({\mathrm{DX}}^{e}=\frac{\mathrm{L.}{\sigma }_{y}}{E}={0.7510}^{-3}m\)

\(\mathrm{DX}\) varies from \({\mathrm{DX}}^{e}\) to \({\mathrm{3DX}}^{e}\)

Rotation imposed in \(B\)

\({\mathrm{DRZ}}^{e}={0.7510}^{-2}m\)

\(\mathrm{DRZ}\) varies from \({\mathrm{DRZ}}^{e}\) to \(20\mathrm{\times }{\mathit{DRZ}}^{e}\) and then decreases to \(\mathrm{-}2\mathrm{\times }{\mathit{DRZ}}^{e}\)

Note:

In pure bending, \(\mathrm{MZ}\) and \(\mathrm{DRZ}\) do not depend on \(x\). The curvature \(\varphi =\frac{d(\mathrm{DRZ})}{\mathrm{dx}}=\mathrm{DRZ}(B)\)